# On radial behaviour and balanced Bloch functions

### Juan J. Donaire

Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain### Christian Pommerenke

Technische Universität Berlin, Germany

## Abstract

A Bloch function $g$ is a function analytic in the unit disk such that $(1–|z|^2)|g' (z)|$ is bounded. First we generalize the theorem of Rohde that, for every "bad" Bloch function, $g(r \zeta) (r \longrightarrow 1)$ follows any prescribed curve at a bounded distance for $\zeta$ in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that $|g'(z)|$ does not vary much on each circle $\lbrace |z| = r\rbrace$ except for small exceptional arcs. We show e.g. that

$\int^1_0|g'(r \zeta)|dr< \infty$

holds either for all $\zeta \in \mathbb T$ or for none.

## Cite this article

Juan J. Donaire, Christian Pommerenke, On radial behaviour and balanced Bloch functions. Rev. Mat. Iberoam. 15 (1999), no. 3, pp. 429–449

DOI 10.4171/RMI/261